Probability and Statistics for Engineers (STAT 301& 305) - Taibah University

TAIBAH UNIVERSITY ‫جامعة طيبة‬ Faculty of Science ‫كلية العلوم‬ Department of Math. ‫قسم الرياضيات‬ Probability and Statistics for Engineers STAT 301& 305 First Semester 1445 H Teacher :  Lesson 21 Mean...

TAIBAH UNIVERSITY ‫جامعة طيبة‬ Faculty of Science ‫كلية العلوم‬ Department of Math. ‫قسم الرياضيات‬ Probability and Statistics for Engineers STAT 301& 305 First Semester 1445 H Teacher :  Lesson 21 Means and Variances of Linear Combinations of Random Variables Linear Combination of random variables If X1, X2, …, Xn are n random variables and a1, a2, …, an are constants, then the random variable : n Y =  ai X i = a1 X 1 + a2 X 2 +  + an X n i =1 is called a linear combination of the random variables X1,X2,…,Xn. 3 Linear Combination of random variables Theorem 1: If 𝑋 is a random variable with mean 𝜇𝑋 = 𝐸(𝑋), and if 𝑎 and 𝑏 are constants, then: 𝑬(𝒂 𝑿  𝒃) = 𝒂 𝑬(𝑿)  𝒃  𝒂 𝑿 𝒃 = 𝒂 𝑿 ± 𝒃 4 Linear Combination of random variables Corollary 1: Setting 𝑎 = 0 in Theorem 4.5 , we see that 𝐸(𝑏) = 𝑏. Corollary 2: Setting 𝑏 = 0 in Theorem 4.5 , we see that 𝐸(𝑎 𝑋) = 𝑎 𝐸(𝑋). 5 Linear Combination of r.v. (Example 1) Let 𝑋 be a random variable with the following probability density function: 1 2  x ; −1 x  2 f ( x) =  3 0 ; elsewhere Find 𝐸(4𝑋 + 3). 6  Linear Combination of r.v. (Example 1) Solution:  2 1 2 1 2 μ = E(X) =  x f ( x) dx =  x [ x ] dx =  x 3 dx = − −1 3 3 −1 1 1 4 x = 2   x  = 5/4 3 4 x = −1 𝐸(4𝑋 + 3) = 4 𝐸(𝑋) + 3 5 = 4 + 3=8 4 7  Linear Combination of r.v. (Example 1) Another solution:  E[g(X)] =  g ( x) f ( x) dx ; g(X) = 4X+3 −  E(4X+3) =  (4 x + 3) f ( x) dx − 2 1 2 =  (4 x + 3) [ x ] dx =  = 8 −1 3 8 Linear Combination of random variables Theorem 2: If 𝑋1, 𝑋2, … , 𝑋𝑛 are 𝑛 random variables and 𝑎1, 𝑎2, … , 𝑎𝑛 are constants, then: 𝐸 𝑎1 𝑋1 + 𝑎2 𝑋2 + … + 𝑎𝑛 𝑋𝑛 = 𝑎1𝐸 𝑋1 + 𝑎2𝐸 𝑋2 + ⋯ + 𝑎𝑛 𝐸(𝑋𝑛 )  n n E (  ai X i ) =  ai E ( X i ) i =1 i =1 9 Linear Combination of random variables Corollary : If 𝑋, and 𝑌 are random variables, then: 𝐸(𝑋 ± 𝑌) = 𝐸(𝑋) ± 𝐸(𝑌) 10  Linear Combination of random variables Theorem 3 : If 𝑋 is a random variable with variance Var ( X ) =  X2 and if 𝑎 and 𝑏 are constants, then: Var(aX  b) = a Var( X ) 2  aX + b 2 =a X 22 11 Linear Combination of random variables Corollary 1: Setting 𝑎 = 1 in Theorem 4.9 , we see that  2 X +b = 2 X Corollary 2: Setting 𝑏 = 0 in Theorem 4.9 , we see that  2 aX =a  2 2 X 12 Linear Combination of random variables Theorem 4: If 𝑋1, 𝑋2, … , 𝑋𝑛 are 𝑛 independent random variables and 𝑎1, 𝑎2, … , 𝑎𝑛 are constants, then: Var (a1 X 1 + a2 X 2 +..... + an X n ) = a12Var ( X 1 ) + a22Var ( X 2 ) +... + an2Var ( X n )  n n Var (  ai X i ) =  ai Var ( X i ) 2 i =1 i =1 σ 2 a1 X 1 + a2 X 2 ++ an X n =a σ +a σ 2 1 2 X1 2 2 2 X2 ++ a σ 2 n 2 Xn 13 Linear Combination of random variables Corollary : If 𝑋, and 𝑌 are independent random variables, then:· Var (aX + bY ) = a Var (X ) + b Var (Y ) 2 2 Var (aX − bY ) = a Var (X ) + b Var (Y ) 2 2 Var (aX  bY ) = a Var (X ) + b Var (Y ) 2 2 14 Linear Combination of r.v. (Example 2) If 𝑋, and 𝑌 are independent random variables, with σ 2 X = 2., σ 2 Y =4 Find the variance of the random variable Z = 3X − 4Y + 8 15 Linear Combination of r.v. (Example 2) Solution: Var (Z ) = Var (3 X − 4Y + 8) = 9Var ( X ) + 16Var (Y ) + 0 = 9  2 + 16  4 = 82 16 Linear Combination of r.v. (Example 3) If X, and Y are independent random variables, with.  X = 2 , σ = 4 , Y = 7 , σ = 1 2 X 2 Y Find: 1. E (3 X + 7 ) Var (3 X + 7 ) 2. E (5 X + 2Y − 2 ) Var (5 X + 2Y − 2 ) 17 Linear Combination of r.v. (Example 3) Solution: 1. E (3 X + 7 ) = 3E ( X ) + 7 = 3  2 + 7 = 13 Var(3 X + 7 ) = (3) Var( X ) = 9  4 = 36 2 2. E (5 X + 2Y − 2 ) = 5E ( X ) + 2 E (Y ) − 2 = 5  2 + 2  7 − 2 = 22 Var (5 X + 2Y − 2 ) = (5) Var ( X ) + (2 ) Var (Y ) 2 2 = 25  4 + 4  1 = 104 18 Linear Combination of Random Variables Theorem 4.8: Let X, and Y be two independent random variables, then: 𝐸 𝑋𝑌 = 𝐸 𝑋 𝐸 𝑌 Example : If X, and Y are independent random variables with 𝜇𝑋 = 2 and 𝜇𝑌 = 7 , Find the following 𝐸(𝑋𝑌 + 2𝑋 − 3𝑌) ? Solution: 𝐸 𝑋𝑌 + 2𝑋 − 3𝑌 = E X E Y + 2E X − 3E Y = 2 7 + 2 2 − 3 7 = −3 19 Exercise: Q1: Given the following probability distribution (mass) function for the discrete random variable X. X 0 1 2 3 4 𝑇𝑜𝑡𝑎𝑙 𝑓(𝑥) 0.41 0.37 0.16 0.05 0.01 1 1- Find the Cumulative Distribution Function ? 2- Find the expected value of X? 3-Find the variance of X? 4- Fin P(X=6) ? 5- Find P ( X 2 22 Q4: If X is a random variable has the cumulative distribution 0 ; 𝑥2 1- Find 𝑓(𝑥) ? 1 2- Find the value of 𝑎 if 𝑃 𝑋 ≤ 𝑎 = ? 4 23